Subgroups — #88590

Order 432.

71 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1088	C₂	1	1	yes	maximal, characteristic
2	#6199	C₂	1	1	yes	maximal, characteristic
3	#6199	C₂	1	1	yes	maximal, characteristic
4	C₂×#272	C₂²	1	1	yes	characteristic
5	#1088	C₄	1	2	yes	complemented
6	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
7	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
8	D₃×Dic₃	—	6	1	no	—
9	D₃×Dic₃	—	6	1	no	—
10	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
11	#272	C₂×C₄	1	1	yes	characteristic
12	#272	Q₈	1	1	yes	characteristic
13	C₄⋊Dic₃	—	9	1	no	maximal
14	C₂×C₃×D₃	—	6	1	no	—
15	C₂×C₃×D₃	—	6	1	no	—
16	C₃⋊Dic₃	—	2	1	no	—
17	C₃⋊Dic₃	—	2	1	no	—
18	C₃×Dic₃	—	6	1	no	—
19	C₃×Dic₃	—	6	1	no	—
20	H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃×C₄	—	9	1	no	—
22	C₂×Dic₃	—	9	1	no	—
23	C₂×Dic₃	—	9	1	no	—
24	C₄×D₃	—	18	1	no	—
25	C₄×D₃	—	18	1	no	—
26	C₂×C₃²	—	2	1	no	—
27	C₂×C₃²	—	2	1	no	—
28	C₃×D₃	—	6	1	no	—
29	C₃×D₃	—	6	1	no	—
30	C₃×D₃	—	6	1	no	—
31	C₃×D₃	—	6	1	no	—
32	C₄⋊C₄	—	27	1	no	Sylow
33	C₂²×C₃	—	9	1	no	—
34	C₂×D₃	—	6	1	no	—
35	C₂×D₃	—	6	1	no	—
36	C₃×C₄	—	9	2	no	—
37	C₃×C₄	—	18	1	no	—
38	C₃×C₄	—	18	1	no	—
39	Dic₃	—	6	1	no	—
40	Dic₃	—	6	1	no	—
41	Dic₃	—	18	1	no	—
42	Dic₃	—	36	1	no	—
43	C₃²	—	2	1	no	—
44	C₃²	—	2	1	no	—
45	C₂×C₄	—	9	1	no	—
46	C₂×C₄	—	27	1	no	—
47	C₂×C₄	—	27	1	no	—
48	C₂×C₃	—	9	1	no	—
49	C₂×C₃	—	9	1	no	—
50	C₂×C₃	—	12	1	no	—
51	C₂×C₃	—	12	1	no	—
52	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
53	D₃	—	6	1	no	—
54	D₃	—	6	1	no	—
55	D₃	—	6	1	no	—
56	D₃	—	6	1	no	—
57	C₂²	—	9	1	no	—
58	C₄	—	9	2	no	—
59	C₄	—	18	1	no	—
60	C₄	—	18	1	no	—
61	C₃	—	12	1	no	—
62	C₃	—	12	1	no	—
63	C₃	#3773	1	1	yes	characteristic
64	C₂	—	9	1	no	—
65	C₂	—	9	1	no	—
66	C₂	#1088⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#1088⋊C₂
commutator	#272	C₂×C₄
Frattini, socle	C₂×C₃	GO⁺(4,𝔽₂)
Fitting	C₂×H(3,𝔽₃)	D₄
prime radical, cosocle	C₂×#272	C₂²
hypocenter	H(3,𝔽₃)	C₄⋊C₄
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #1088	C₄	semidirect
20. H(3,𝔽₃)	C₄⋊C₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#1088	2
#6199	2
C₄⋊Dic₃	9